Minimal Weierstrass equation
\(y^2=x^3-41386827x+64425512954\)
Mordell-Weil group structure
$\Z\times \Z/{2}\Z$
Infinite order Mordell-Weil generator and height
$P$ | = |
\(\left(-697, 304850\right)\)
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$\hat{h}(P)$ | ≈ | $5.2813532302471566600084353313$ |
Torsion generators
\( \left(1669, 0\right) \)
Integral points
\((-697,\pm 304850)\), \( \left(1669, 0\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 115920 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | $2743895437362384076800000 $ | = | $2^{42} \cdot 3^{11} \cdot 5^{5} \cdot 7^{2} \cdot 23 $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{2625564132023811051529}{918925030195200000} \) | = | $2^{-30} \cdot 3^{-5} \cdot 5^{-5} \cdot 7^{-2} \cdot 23^{-1} \cdot 1979^{3} \cdot 6971^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $3.3907035350110211489708101835\dots$ | ||
Stable Faltings height: | $2.1482502101170209938559554436\dots$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | $5.2813532302471566600084353313\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | $0.074122273445691298079878501637\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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Tamagawa product: | $ 80 $ = $ 2^{2}\cdot2\cdot5\cdot2\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | $2$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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Special value: | $ L'(E,1) $ ≈ $ 7.8293181659132956046825624237 $ |
Modular invariants
Modular form 115920.2.a.ep
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 13824000 | ||
$ \Gamma_0(N) $-optimal: | yes | ||
Manin constant: | 1 |
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{34}^{*}$ | Additive | -1 | 4 | 42 | 30 |
$3$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 2 | 11 | 5 |
$5$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$23$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
---|---|---|
$2$ | 2B | 2.3.0.1 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | split | split | ss | ord | ss | ss | nonsplit | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | - | - | 2 | 6 | 1,1 | 1 | 1,3 | 1,1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 |
$\mu$-invariant(s) | - | - | 0 | 0 | 0,0 | 0 | 0,0 | 0,0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 115920.ep
consists of 2 curves linked by isogenies of
degree 2.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{345}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
$4$ | 4.0.4327680.7 | \(\Z/4\Z\) | Not in database |
$8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
$8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
$8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.